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An effective version of the Lindemann-Weierstrass theorem for interpolation determinants. (Une version effective du théorème de Lindemann-Weierstrass par les déterminants d’interpolation.) (French) Zbl 0935.11024

The best known quantitative version of the Lindemann-Weierstrass theorem was given by M. Ably [Acta Arith. 67, 29-45 (1994; Zbl 0813.11042)] by using Philippon’s criterion for algebraic independence. Here the author uses Laurent’s interpolation determinants to obtain a lower bound of the same type for \(|P(e^{\alpha_1},\dots, e^{\alpha_p})|\) where the \(\alpha_j\) are complex algebraic numbers, linearly independent over \(\mathbb{Q}\), and \(P\) is a polynomial with algebraic coefficients. In comparison with Ably’s result (where \(P\) has rational coefficients) this bound is a little better for the dependence on \(P\) and especially entirely explicit for the dependence on the \(\alpha_j\). This paper is a very nice piece of virtuosity.

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J82 Measures of irrationality and of transcendence

Citations:

Zbl 0813.11042
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References:

[1] Ably, M., Une version quantitative du théorème de Lindemann-Weierstrass, Acta Arith., 67, 29-30 (1994) · Zbl 0813.11042
[2] Bombièri, E.; Vaaler, J., On Siegel’s Lemma, Inv. Math., 73, 11-33 (1983) · Zbl 0533.10030
[3] Gramain, E., Sur le lemme de Siegel (d’après E. Bombièri et J. Vaaler), Publ. Math. Paris VI, 2.1-2.37 (1984)
[4] Laurent, M., Linear forms in two logarithms and interpolation determinants, Acta Arith. (1994) · Zbl 0801.11034
[5] Mahler, K., Zur Approximation der Exponential Funktion und des Logarithmus I, J. Reine Angew. Math., 166, 118-136 (1932) · JFM 57.0242.03
[6] Waldschmidt, M., Report (1992)
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